The objective of the present invention is to determine the presence of signals in the frequency spectrum and characterize them. These signals are typically represented in terms of parameters such as: energy, noise, signal-to-noise ratio (SNR), bandwidth and center frequency.
The prior art has provided spectral estimation techniques where the receiver collects a wide band signal that contains frequency allocated signals in different formats. The wideband signal may contain tones and analog and digital modulation types. Spectral survey estimation methods do a spectral search to identify signals of interests. Thus far, prior art method for the detection of signals requires the use of frequency or time domain techniques.
Referring to FIG. 1, a prior art spectral survey estimator is depicted. The analysis filter bank 10 decomposes the input signal in narrow band channels to produce a spectral estimate. This step can also be accomplished by using DFT based spectral methods. The output of the channels are sent to a spectral survey estimator 20 such as the one presented in [3]. The process estimates the number of signals that are present and determines the corresponding frequency and bandwidth estimates. The estimates are used to recover the individual signals. FIG. 2 depicts the flow of the entire process corresponding to the hardware implementation of FIG. 1.
Prior art methods [1, 2] use time domain techniques that require the use of frequency locking devices. These methods are more suitable for a hardware implementation because phase locking techniques are inefficient when applied for digitized samples [17].
Developments in speech processing, for example, [7] have provided some methodology for classifying speech signal spectra. Basu and other authors acknowledge that a special case of the expectation maximization (EM) method known as Gaussian Mixture Models does not always provide a suitable statistical model. In the case of speech processing, it is important to model heavy tailed distributions. Basu also uses several models of non-Gaussian distributions. This particular distribution has a flatter response and provides the convenience of reducing the complexity of the expectation when the log-likelihood is applied. Basu's method is applied on cepstral vectors which are used for extracting formant and vocal track information. There is no implicit or explicit intention to detect and demultiplex signals.
Referring to FIG. 3, prior art Gaussian Mixture Models (GMM) are a special case of the maximization of expectation process. It defines the model in terms of Gaussian densities. The parameters of the model include: the probability of each mixture element, a vector of the mean and a covariance matrix. While the use of a Gaussian distribution simplifies the maximization of the log-likelihood and produces a matrix equation that is easy to implement, it does not produce acceptable results.
Failings of the Gaussian Mixture Approach for Signal Detection
A frequency spectrum can be treated in the same manner as a mixture of probability density functions. Because the profile of communications signals in the spectrum resembles bell-shape densities (see FIG. 3), a reasonable first choice would be to model signals using Gaussian distributions. In a one-dimensional space, the mixture probability α becomes the amplitude-bandwidth product; the mean μ becomes the center frequency f; and the standard deviation σ, the bandwidth b, the data vector {right arrow over (y)}′ becomes a scalar y′. The data contains the information about the frequencies in the spectrum.
                              θ          →                =                              [                                                                                θ                    1                                                                                        θ                    2                                                                                        θ                    3                                                                        ]                    =                                    [                                                                    α                                                        μ                                                        σ                                                              ]                        →                          [                                                                    α                                                        f                                                        b                                                              ]                                                          Eq        .                                  ⁢        1                                                      p            (                                                            y                  ′                                |                                  z                  ′                                            ;                              θ                →                                      )                    ·                      P            (                                          z                ′                            ;                              θ                →                                      )                          =                              1                                                            2                  ⁢                                                                          ⁢                  π                                            ⁢                              θ                3                                              ⁢                                    exp              (                              -                                                                            (                                                                        y                          ′                                                -                                                  θ                          2                                                                    )                                        2                                                        2                    ⁢                                                                                  ⁢                                          θ                      3                      2                                                                                  )                        ·                          θ              1                                                          Eq        .                                  ⁢        2            The GMM must be adapted to process histograms, or spectral data in this specific case. The approach will consider two sets. The first set describes J′ statistical events that represent our known data.Y′={y′j:j=1:J′}  Eq. 3A second set is defined as the histogram of the set Y′. The bin number (frequency) and the count or weight (amplitude) are represented by the pair yj and wj respectively.Y={(yj,wj):yj=bin range,j=1:J,wj=count}  Eq. 4Associated with each bin yj, there is also a quantity zj that contains information about the mixture element that corresponds to the bin. The mixture probability or the probability of finding a mixture element k should not be affected by the change of variables as long as both variables z and z′ are referring to the same mixture element.P(z′=k;{right arrow over (θ)})=P(z=k;{right arrow over (θ)})  Eq. 5The probability of the data y′j given a mixture element z′j must be expressed in terms of the new variables yj, wj and zj. For this purpose, we define a new conditional “probability” {circumflex over (p)}(yj|zj=k;{right arrow over (θ)}) as an average of the probabilities of all the statistical samples y′j inside the bin yj. It is assumed that events inside a bin share the same variable z.
                                          p            ^                    (                                                                      y                  j                                |                                  z                  j                                            =              k                        ;                          θ              →                                )                ≡                              1                          w              j                                ⁢                                    ∑              m                        ⁢                          p              (                                                                                                                  y                        m                        ′                                            ⊆                                              y                        j                                                              |                                          z                      j                      ′                                                        =                  k                                ;                                  θ                  →                                            )                                                          Eq        .                                  ⁢        6            The probability density of the data within a bin given by:
                                          ∑            m                    ⁢                      p            (                                                            y                  m                  ′                                ⊆                                  y                  j                                            ,                                                                    z                    j                                    =                  k                                ;                                  θ                  →                                                      )                          =                              w            j                    ·                                    p              ^                        (                                                                                y                    j                                    |                                      z                    j                                                  =                k                            ;                              θ                →                                      )                    ·                      P            (                                                            z                  j                                =                i                            ;                              θ                →                                      )                                              Eq        .                                  ⁢        7                                                      ∑            m                    ⁢                      p            (                                                            y                  m                  ′                                ⊆                                  y                  j                                            ;                              θ                →                                      )                          =                              ∑                          k              =              1                        K                    ⁢                                    w              j                        ·                                          p                ^                            (                                                                                          y                      j                                        |                                          z                      j                                                        =                  k                                ;                                  θ                  →                                            )                        ·                          P              (                                                                    z                    j                                    =                  k                                ;                                  θ                  →                                            )                                                          Eq        .                                  ⁢        8            The new a posteriori probability would take the form of equation 9. The term wj is cancelled and this leaves an equation that is similar to the original Bayesian estimate.
                                                        ∑              m                        ⁢                          p              (                                                                    z                    j                                    =                                      i                    |                                                                  y                        m                        ′                                            ⊆                                              y                        j                                                                                            ;                                                      θ                    →                                    c                                            )                                =                                                                      p                  ^                                (                                                                                                    y                        j                                            |                                              z                        j                                                              =                    i                                    ;                                                            θ                      →                                        c                                                  )                            ⁢                              P                (                                                                            z                      j                                        =                    i                                    ;                                                            θ                      →                                        c                                                  )                                                                                                                                ∑                                              k                        =                        1                                            K                                        ⁢                                                                                            p                          ^                                                (                                                                                                                                            y                                j                                                            |                                                              z                                j                                                                                      =                            k                                                    ;                                                                                    θ                              →                                                        c                                                                          )                                            ·                                                                                                                                        P                    (                                                                                            z                          j                                                =                        k                                            ;                                                                        θ                          →                                                c                                                              )                                                                                      ⁢                                  ⁢                              p            ⁡                          (                                                                    z                    j                                    =                                      i                    |                                          y                      j                                                                      ;                                                      θ                    →                                    c                                            )                                ≡                                    ∑              m                        ⁢                          p              ⁡                              (                                                                            z                      j                                        =                                          i                      |                                                                        y                          m                          ′                                                ⊆                                                  y                          j                                                                                                      ;                                                            θ                      →                                        c                                                  )                                                    ⁢                                  ⁢                              With            ⁢                                                  ⁢            K                    =                      number            ⁢                                                  ⁢            of            ⁢                                                  ⁢            mixture            ⁢                                                  ⁢                          elements              .                                                          Eq        .                                  ⁢        9            The histogram approach can be incorporated into equation 12 as follows:
                              Q          ⁡                      (                          θ              →                        )                          =                              E            z                    ⁢                      {                                          ln                ⁡                                  (                                                            ∏                                              j                        =                        1                                            J                                        ⁢                                                                                  ⁢                                                                                            p                          ⁡                                                      (                                                                                                                                                                y                                    →                                                                    j                                                                |                                                                  z                                  →                                                                                            ;                                                              θ                                →                                                                                      )                                                                                                    w                          j                                                                    ·                                                                        P                          ⁡                                                      (                                                                                          z                                →                                                            ;                                                              θ                                →                                                                                      )                                                                                                    w                          j                                                                                                      )                                            |                              y                →                                      }                                              Eq        .                                  ⁢        10            
It has been assumed that the weight (amplitude) wj is a positive integer corresponding to the number of repetitions. The proposed process extends the definition of the weights for any positive number. The pair (yj,wj) will be used to represent the spectral density with frequency bin yj and amplitude wj.
The Gaussian-based process was developed using similar steps as those found in [21] and incorporated the modifications of equations 6 through 10. Then the expression was maximized and the formulas for updating the parameters were obtained.
After running several simulations, it was observed that the foregoing Gaussian-based process fails to identify signal parameters or useful features for estimation of the signal parameters. For a Gaussian model, there is no sharp transition between the pass band and the stop band. These wide transitions interfere with ones of the adjacent signals. The process has convergence problems due to the interference between Gaussian mixture elements and the model is not suitable for characterizing communication signals. A new process with a flatter pass band and clear transition between the pass and stop band is needed (see FIG. 6) and provides the motivation for the present invention.
Non-Gaussian Approaches for Spectral Survey
Other probability densities were explored to overcome the limitations and failings of Gaussian-based processes for the estimation of signal parameters. First, a Butterworth filter model was used [Eq. 11]. But this process is not convenient because it produces a combination of complex logarithmic and rational expressions when the expectation is maximized.
                              p          ⁡                      (                          y              ,                              z                |                                  θ                  →                                                      )                          =                                            p              ⁡                              (                                                      y                    |                    z                                    ;                                      θ                    →                                                  )                                      ·                          P              ⁡                              (                                  z                  ;                                      θ                    →                                                  )                                              =                                    1                              1                +                                                      (                                                                  y                        -                        f                                            b                                        )                                                        2                    ⁢                                                                                  ⁢                    N                                                                        ·            α                                              Eq        .                                  ⁢        11            